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18
A REVIEW OF PARRONDO’S PARADOX
, 2002
"... Inspired by the flashing Brownian ratchet, Parrondo’s games present an apparently paradoxical situation. The games can be realized as coin tossing events. Game A uses a single biased coin while game B uses two biased coins and has a state dependent rule based on the player’s current capital. Playin ..."
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Cited by 29 (5 self)
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Inspired by the flashing Brownian ratchet, Parrondo’s games present an apparently paradoxical situation. The games can be realized as coin tossing events. Game A uses a single biased coin while game B uses two biased coins and has a state dependent rule based on the player’s current capital. Playing each of the games individually causes the player to lose. However, a winning expectation is produced when randomly mixing games A and B. This phenomenon is investigated and mathematically analyzed to give explanations on how such a process is possible. The games are expanded to become dependent on other properties rather than the capital of the player. Some of the latest developments in Parrondian ratchet or discretetime ratchet theory are briefly reviewed.
Capital Redistribution Brings Wealth By Parrondo's Paradox
, 2002
"... this paper we introduce a new scenario for the Parrondo's paradox which involves a set of players ~ [7] and where one of the games has been replaced by a redistribution of the capital owned by the players. It will be shown that even though each individual player (when playing alone) has a neg ..."
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Cited by 15 (6 self)
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this paper we introduce a new scenario for the Parrondo's paradox which involves a set of players ~ [7] and where one of the games has been replaced by a redistribution of the capital owned by the players. It will be shown that even though each individual player (when playing alone) has a negative winning expectancy, the redistribution of money brings each player a positive expected gain. This result holds even in the case that the redistribution of capital is directed from the richer to the poorer, although in this case the distribution of money amongst the players is more uniform and the total gain is less
Discrete–time ratchets, the FokkerPlanck equation and Parrondo’s paradox
 Accepted in Proc. R. Soc. London A. Proc. of SPIE
, 2004
"... Parrondo’s games manifest the apparent paradox where losing strategies can be combined to win and have generated significant multidisciplinary interest in the literature. Here we review two recent approaches, based on the FokkerPlanck equation, that rigorously establish the connection between Parro ..."
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Cited by 8 (5 self)
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Parrondo’s games manifest the apparent paradox where losing strategies can be combined to win and have generated significant multidisciplinary interest in the literature. Here we review two recent approaches, based on the FokkerPlanck equation, that rigorously establish the connection between Parrondo’s games and a physical model known as the flashing Brownian ratchet. This gives rise to a new set of Parrondo’s games, of which the original games are a special case. For the first time, we perform a complete analysis of the new games via a discretetime Markov chain (DTMC) analysis, producing winning rate equations and an exploration of the parameter space where the paradoxical behaviour occurs. Keywords: Parrondo’s paradox; FokkerPlanck equation; Brownian ratchet. 1.
Limit theorems and absorption problems for onedimensional correlated random walks
, 2003
"... In this paper we consider limit theorems and absorption problems for correlated random walks determined by a 2 × 2 transition matrix on the line by using a basis P,Q,R,S of the vector space of real 2 × 2 matrices as in the case of our analysis on quantum walks. ..."
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Cited by 7 (3 self)
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In this paper we consider limit theorems and absorption problems for correlated random walks determined by a 2 × 2 transition matrix on the line by using a basis P,Q,R,S of the vector space of real 2 × 2 matrices as in the case of our analysis on quantum walks.
2002a Quantum coherence, correlated noise and Parrondo games. Fluct
 Noise Lett
, 2002
"... We discuss the effect of correlated noise on the robustness of quantum coherent phenomena. First we consider a simple, toy model to illustrate the effect of such correlations on the decoherence process. Then we show how decoherence rates can be suppressed using a Parrondolike effect. Finally, we re ..."
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Cited by 6 (0 self)
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We discuss the effect of correlated noise on the robustness of quantum coherent phenomena. First we consider a simple, toy model to illustrate the effect of such correlations on the decoherence process. Then we show how decoherence rates can be suppressed using a Parrondolike effect. Finally, we report the results of manybody calculations in which an experimentallymeasurable quantum coherence phenomenon is significantly enhanced by nonMarkovian dynamics arising from the noise source. 1
Order from disorder: The role of noise in creative processes: A special issue on game theory and evolutionary processes — overview. Fluctuation and Noise Letters 2
, 2002
"... importance of applying game theory to the evolution of information in the presence of noise has recently become widely recognized. This Special Issue addresses the theme of spontaneously emergent order in both classical and quantum systems subject to external noise, and includes papers directly rel ..."
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Cited by 5 (4 self)
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importance of applying game theory to the evolution of information in the presence of noise has recently become widely recognized. This Special Issue addresses the theme of spontaneously emergent order in both classical and quantum systems subject to external noise, and includes papers directly related to game theory or the development of supporting techniques. In the following editorial overview we examine the broader context of the subject, including the tension between the destructive and creative aspects of noise, and foreshadow the significance of some of the subsequent papers in the volume.
Quantum Walks
, 2008
"... Abstract. Quantum walks can be considered as a generalized version of the classical random walk. There are two classes of quantum walks, that is, the discretetime (or coined) and the continuoustime quantum walks. This manuscript treats the discrete case in Part I and continuous case in Part II, re ..."
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Cited by 4 (0 self)
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Abstract. Quantum walks can be considered as a generalized version of the classical random walk. There are two classes of quantum walks, that is, the discretetime (or coined) and the continuoustime quantum walks. This manuscript treats the discrete case in Part I and continuous case in Part II, respectively. Most of the contents are based on our results. Furthermore, papers on quantum walks are listed in References. Studies of discretetime walks appeared from the late 1980s from Gudder (1988), for example. Meyer (1996) investigated the model as a quantum lattice gas automaton. Nayak and Vishwanath (2000) and Ambainis et al. (2001) studied intensively the behaviour of discretetime walks, in particular, the Hadamard walk. In contrast with the central limit theorem for the classical random walks, Konno (2002a, 2005a) showed a new type of weak limit theorems for the onedimensional lattice. Grimmett, Janson, and Scudo (2004) extended the limit theorem to a wider range of the walks. On the other hand, the continuoustime quantum walk was introduced and studied by Childs, Farhi, and Gutmann (2002). Excellent reviews on quantum walks are
www.elsevier.com/locate/physa Quantum models of Parrondo’s games
"... A Parrondo’s paradox is an e*ect where two losing games, when combined, can produce a net winning result. We provide a short introduction to quantum versions of Parrondo’s games and review the current status of the work. ..."
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Cited by 1 (0 self)
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A Parrondo’s paradox is an e*ect where two losing games, when combined, can produce a net winning result. We provide a short introduction to quantum versions of Parrondo’s games and review the current status of the work.
Simulation of a Quantum Random Walk
"... For my project I simulated a quantum random walk in one plus one dimensions with a ratcheting potential applied. The particle begins at the origin and after running the simulation for one hundred time steps the most probable final position was calculated and then graphed according to its initial con ..."
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For my project I simulated a quantum random walk in one plus one dimensions with a ratcheting potential applied. The particle begins at the origin and after running the simulation for one hundred time steps the most probable final position was calculated and then graphed according to its initial condition. These results were analyzed to determine the dependence on initial condition of final position.